Gyrokinetic simulations of ETG turbulence and Gyrokinetic - - PowerPoint PPT Presentation

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Gyrokinetic simulations of ETG turbulence and Gyrokinetic simulations of ETG turbulence and zonal flows in positive/reversed shear tokamaks zonal flows in positive/reversed shear tokamaks

Yasuhiro Idomura Yasuhiro Idomura Japan Atomic Energy Research Institute Festival de Festival de Theorie Theorie 2005 2005 Aix Aix-

• en

en-

• Provance

Provance, France, 4 , France, 4-

• 22 July 2005

22 July 2005 Outline

• Gyrokinetic simulations of toroidal ETG turbulence
• Linear and quasi-linear analysis of ETG mode
• ETG turbulence simulation in PS/RS tokamaks
• Self-organization in ETG turbulence

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Motivation to study ETG turbulence Motivation to study ETG turbulence

ETG turbulence is experimentally relevant candidate of χe in tokamak

– High suppression threshold ωExB > γ than TEM (Stallard 1999) – Stiff Te profile consistent with critical Lte of ETG (Hoang 2001)

Issues to be addressed

Does ρe scale ETG turbulence cause experimentaly relevant χe?

– Yes: χe~10χGB (χGB=vte ρ te

2/Lte) in ρ*-1~∞ (ρ*-1=a/ρ te) local flux

tube toroidal GK code (Jenko 2002)

– No: χe~χGB in ρ*-1~100 global toroidal GF code (Labit 2003)

What kind of structure formations does ETG turbulence show?

– Streamers: positive shear flux tube toroidal GK code (Jenko 2002) – Zonal flows: reversed shear global slab GK code (Idomura 2000)

To examine these qualitatively and quantitatively different results,

ETG turbulence is studied using global toroidal GK simulations

– ρ*-dependence of toroidal ETG modes – Zonal flow and streamer formations in PS/RS-ETG turbulence

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Basic equations Basic equations

Electrostatic gyrokinetic equation (Hahm 1988) Gyrokinetic Poisson equation

{ } { }

( )

{ }

( )

B v m q v cm B B v m q B m H v dt dv B B v m q B q c v H dt d H F t F Dt DF q B v m H

e e e e g e e e g e e e e e g e e

2 , ln , ln , , 2 1

2 // * 2 // * // * // // 2 // * // // 2 // ⊥

= × ∇ + = ∇ + ∇ ⋅ + ∇ ⋅ − = ≡ ∇ + ∇ ⋅ + ∇ × + = ≡ = + ∂ ∂ = + + = µ µ φ µ φ φ µ b B B b b B b b b b R R

R R R R R R R

[ ] ( )

Z x ρ R

6 * // 2 2 2 2 2

4 1 d B m F q

e e e e Di De te

∫

− + = + ∇ ⋅ ∇ − ∇ −

⊥ ⊥

δ δ π φ λ φ λ ρ φ

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• Electrostatic GK toroidal PIC code
• Gyrokinetic electrons with adiabatic ions (k⊥ρti>>1)
• Annular wedge torus geometry

–

fixed B.C. φ = 0

–

n = 0, N, 2N… (N=25~100)

• Quasi-ballooning representation
• Global profile effects (ne, Te, q, 1/r)

–

Self-consistent Te, ne are relaxed by heat/particle transport

–

ω*

te-shearing effect

–

Reversed q(r) profile

• Optimized particle loading

–

energy/particle conservation

Calculation models of ETG turbulence simulation Calculation models of ETG turbulence simulation

Validity of simulation is checked by conservation properties !

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High High-

• n

n solver with quasi solver with quasi-

• ballooning representation

ballooning representation

Realistic tokamak size a/ρte~104: kθρte~1 (q=1.4) m=5000

–

~104 poloidal grids are needed without QB representation

–

~102 poloidal grids are enough with QB representation

jump condition for periodicity in θ

mode structure

• n the poloidal plane

mode structure along the field line

( ) ( )

( )

( ) ( )

( )

surface reference : , 2 , ˆ , ˆ , , ˆ , ,

ˆ 2 ˆ ˆ s r q n i n n n r q in in n

r e r r e r r

s s

π θ ϕ

π φ φ θ φ ϕ θ φ = = ∑

+ −

φ φ ˆ

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Linear and quasi-linear analysis of ETG mode

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Linear ETG growth rate spectrum Linear ETG growth rate spectrum

Cyclone like parameters (R0/Lte=6.9,ηe=3.12,a~8600ρte~150ρti)

– Unstable region spreads over n~2000 (m~3000, kθρte~0.7) – RS-ETG mode is excited around qmin surface (Idomura 2000) – Almost the same γmax in PS and RS configurations

analysis domain

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Toroidal mode coupling in PS/RS configurations Toroidal mode coupling in PS/RS configurations

Positive shear configuration

– Ballooning PS-ETG mode – Big streamer structure in

weak field side

Reversed shear configuration

– Slab like RS-ETG mode – Single helicity feature in

weak shear region

r safety factor q resonant surfaces positive shear negative shear ballooning mode stable mode resonant perturbations nonresonant slab mode r safety factor q q=m/n q=(m+1)/n q=(m-1)/n q=(m-2)/n q=(m+2)/n resonant surfaces resonant perturbations ballooning mode q=(m-3)/n q=(m-4)/n q=(m+3)/n q=(m+4)/n

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ρ ρ*

* scan of

scan of eigenfunctions eigenfunctions in PS/RS tokamaks in PS/RS tokamaks

Positive shear configuration

– ∆r of PS-ETG mode is limited by ω*-shearing effect (Kim 1994) – ∆r of RS-ETG mode is determined by q profile (Idomura 2000)

Reversed shear configuration

non-resonant resonant ~60ρte ~120ρte a/ρte~536 a/ρte~2146 a/ρte~2146 a/ρte~536 qmin surface ~45ρte ~40ρte

2 / 1 * te

/

−

∝ ∆ ρ ρ r

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Mixing length theory and Mixing length theory and ρ ρ*

*-

• scaling

scaling

Mixing length theory of ETG modes in PS/RS plasmas

– PS-ETG mode – RS-ETG mode

ρ* scan of the saturation amplitude in single-n simulations

– Small ρ* PS-ETG modes give order of magnitude higher

saturation level than RS-ETG and large ρ* PS-ETG modes

Fixed local parameters R0/Lte=6.9, ηe=3.12 kθρte~0.3, a/R0=0.358 γNL: eddy turn over time

2 / 1 * te

/

−

∝ ∆ ρ ρ r

( )

2 / 1 n ns te

/ /

−

∝ ∆ L L r ρ

1 * n GB ML / −

∝ ρ γ χ χ

n ns n GB ML

/ / L L γ χ χ ∝

te n n

/ v L γ γ =

( )

1 n

/ ln

−

= dr n d L

e

( )

2 / 1 ns

/ 2 r q qR L ′ ′ =

L NL

γ γ ≈

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ETG turbulence simulation in PS/RS tokamaks

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Streamer formation in PS Streamer formation in PS-

• ETG turbulence

ETG turbulence

Linear phase (t vte/Ln ~110) Saturation phase (t vte/Ln ~208)

– PS-ETG turbulence is dominated by streamers – Streamers are characterized by ballooning structure and ω~ω*

e

QL streamers (t vte/Ln ~175) Nonlinear streamers (t vte/Ln ~250)

kθρte ~ 0.27 kθρte ~ 0.17 ω ~ ω*

e

weak field side θ ~ 0 600ρte θ r

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χe/(vteρte

2/Lte)

<VExB>/vte R0/Lte

R0/Lte~6.9 R0/Lte~5.5 (R0/Lte)crit~4.5 ~5γ -1 zonal flows χe~10χGB

Te profile is strongly relaxed in a turbulent time scale ~5γ -1

Extremely high Extremely high χ χe in PS in PS-

• ETG turbulence

e

ETG turbulence

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Convergence of saturation levels against wedge size Convergence of saturation levels against wedge size

Time history of fluctuation field energy

– Saturation amplitude is converged against wedge torus size – Does nonlinear toroidal mode coupling (Lin 2004) lower

saturation level?

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Convergence of Convergence of n n-

• spectrum against wedge size

spectrum against wedge size

1/100 wedge torus, 32 mode

– Nonlinear spectrum is converged to coherent streamer mode – QL streamers are excited at linearly most unstable kθρte – 2nd streamers have coherent structure with kθρte~0.2 – Zonal flow component is very small

1/25 wedge torus, 128 mode |φn| (r/a~0.5)

second streamers kθρte~0.17 quasi-linear streamers kθρte~0.27

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Zonal flow formation in RS Zonal flow formation in RS-

• ETG turbulence

ETG turbulence

Linear phase (t vte/Ln ~110) Secondary mode (t vte/Ln ~255)

– RS-ETG turbulence show qualitatively different behavior across qmin – Zonal flows (streamers) appear in negative (positive) shear region

Saturation phase (t vte/Ln ~207) Zonal flow formation (t vte/Ln ~380) qmin

kθρte ~ 0.27

weak field side θ ~ 0 600ρte θ r

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zonal flows

R0/Lte

(R0/Lte)crit~3.7

qmin

χ χe

e gap structure in RS

gap structure in RS-

• ETG turbulence

ETG turbulence

χe/(vteρte

2/Lte)

<VExB>/vte

qmin qmin

Te gradient is sustained above its critical value in quasi-steady state

quasi-steady zonal flows χe suppression

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Summary(1) Summary(1)

ETG turbulence is studied using global toroidal GK simulations Initial saturation levels consistent with the mixing length theory

– Ballooning PS-ETG modes show Bohm like ρ*-scaling – Slab like RS-ETG modes show gyro-Bohm like ρ*-scaling – Small ρ* PS-ETG modes give an order of magnitude higher

saturation level than RS-ETG and large ρ* PS-ETG modes

PS/RS ETG turbulences show different structure formations

– PS-ETG turbulence is dominated by streamers

Te profile is quickly relaxed by large χe~10χGB

– RS-ETG turbulence is characterized by zonal flows

(streamers) in negative (positive) shear region

Te profile is sustained by χe gap structure

These results suggest a stiffness of Te profile in PS tokamaks,

and a possibility of the Te transport barrier in RS tokamaks

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Self-organization in ETG turbulence

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Motivation to study self Motivation to study self-

• organization of ETG
• rganization of ETG-
• ZF

ZF

ZFs are common phenomena not only in drift wave

turbulence but also in Rossby wave turbulence

While ZFs in Rossby wave turbulence are formed

by self-organization (Williams 1978), ITG-ZFs are generated by modulational instability (Chen 2000) →Absence of adiabatic electron response to ITG- ZFs enhances modulational instability (Li 2002)

In contrast, because of adiabatic ion response to

ETG-ZFs, modulational instability is weak in ETG turbulence, and its governing equation is almost the same as Rossby wave turbulence →Does similar generation mechanism exist?

To clarify generation mechanism of ETG-ZFs, self-organization

process is studied in decaying electron turbulence simulations

– Inverse energy cascade – Rhines scale length same? different? ZF on planets ZF in plasmas

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Rhines Rhines scale length in Hasegawa scale length in Hasegawa-

• Mima

Mima Eq Eq. .

In the limit of k//→0, slab GK equations reduce to HM equation Conservation of energy E and potential enstrophy W

– Turbulent binomial cascade (Hasegawa 1978)

→Inverse energy cascade c.f. Selective dissipation of W (Kraichnan 1967)

kr regimes with wave like (linear term) and turbulent (nonlinear

term) features are separated by Rhines scale length (Rhines 1975)

( ) ( )

i e te De s s s

T T n t = + = = + ∇ ∇ ⋅ ∇ × + − ∇ ∂ ∂

⊥ ⊥ ⊥ ⊥

τ ρ λ ρ φ ρ φ τφ φ ρ , 1 ln

2 2 2 2 2 2 2

b

( ) ( ) ( ) dV

U L n L U k

n n s 2 2 / 1 1 4 / 1 2 / 1 4 / 3 2 / 1

2 1 , 2 , ln 2 2 /

∫

⊥ − ⊥ − − −

∇ = = = ∇ = = = φ ε ε β ε β ρ

β

( )

( )

( )

∫ ∫

⊥ ⊥ ⊥

∇ + ∇ = ∇ + = dV W dV E

2 2 2 2 2

2 1 , 2 1 φ φ φ φ

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• Electrostatic GK slab PIC code (2.5D)
• Gyrokinetic electrons with adiabatic ions (k⊥ρti>>1)
• Single helicity shear less slab model for qmin region

–

coordinate system (x,y,z)

–

fixed (φ = 0) and periodic boundary conditions in x and y

–

system size Ly = 292 ρte, Lx = 2 ~ 8Ly

–

k// = B1/|B|ky = 8×10-5ky (kz=0) →E and W decay by Landau damping

• Plasma parameters

–

ρte

2/λDe 2 ~ 0.1, τ ~ 0.3

–

Ln = 183ρte ~ ∞, Lte = ∞

• Initial condition

–

sub-grid random noise

–

eφ/Te ~ 0.005

Decaying electron turbulence simulations Decaying electron turbulence simulations

y B z B x L n

n

∇ + ∇ = ∇ = ∇

− ⊥ 1 1

, ln B Contour plot of φ at t = 0

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Turbulent structure is changed by density gradient Turbulent structure is changed by density gradient

Contour plot of φ in quasi-steady relaxed state (tΩi = 870)

Ln=∞

– Coherent isotropic vortices

are produced

– Merger of like-sign vortices

and decrease of vortices →c.f. 2D fluid turbulence (McWilliams 1984)

Ln=1462ρte

※Lx=2Ly=584ρte

– Anisotropic turbulent

structure with zonal flows

– Zonal flows are produced

by self-organization

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Inverse energy cascade in self Inverse energy cascade in self-

• organization
• rganization

Time history of E and W

※ Lx=8Ly=2336ρte

– W decays much faster than E – Average zonal flow wave number shifts to upscale in time

Time history of

∫ ∫

=

x k x k x x

dk E dk E k k

x x

Decaying electron turbulence simulation with Ln=731ρte

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Rhines Rhines scale length in electron turbulence scale length in electron turbulence

Summary of E, W and

values observed at tΩi = 1000

– Larger Ln

• 1 or ω*/k// leads to fluid limit (small dissipation)

–

becomes larger as Ln

• 1 increases

– Scaling is consistent with Rhines scale length

Scaling of Ln, ε and

4 / 1 2 / 1 − −

∝ ε

n x

L k 0.301 0.136 0.761 183 0.240 0.087 0.634 366 0.188 0.039 0.362 731 0.178 0.012 0.079 1462 ρte W(t)/W(0) E(t)/E(0) Ln/ρte

x

k

x

k

x

k

x

k

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Role of density gradient in ETG turbulence Role of density gradient in ETG turbulence

ηe=∞ (linear phase) ηe=∞ (quasi-steady phase) ηe=5 (linear phase) ηe=5 (quasi-steady phase)

ETG turbulence simulations with ηe=Ln/Lte=∞ and ηe=5

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χ χe

e and ETG zonal flows can be

and ETG zonal flows can be controled controled by by L Ln

n

Time history of χe in ETG turbulence with ηe=∞ and ηe=5

※ Lte is chosen so that the same linear growth rates are given.

– Initial and quasi-steady saturation levels are χe~14χGB

(χe~3χGB) and χe~3χGB (χe~0.7χGB) with ηe=∞ (ηe=5)

– χe is enhanced more than 4 times by flat density profile.

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Summary (2) Summary (2)

Generation mechanism of ETG-ZFs is studied using decaying

electron turbulence simulations

ETG-ZFs are produced by self-organization processes

– Inverse energy cascade is observed – ZF wave number is determined by Rhines scale length

→ETG-ZFs can be controlled by density gradient

Controllability of ETG-ZF is tested in ETG turbulence simulations

with and without density gradient

– Isotropic (anisotropic) turbulent structure is produced without

(with) density gradient

– χe is enhanced more than 4 times by flat density profile

Slab ETG model may explain χe in tokamak core region where

toroidal ETG mode and trapped electron mode are weak